# When did ordered pairs first appear?

I am interested in the history of the ordered pair.

When did ordered pairs as a concrete object, not necessarily defined in terms of sets, appear in a mathematical or logical setting?

The definition of ordered pairs in terms of sets goes back to Norbert Wiener with $$\{\{\{a\}, \varnothing\}, \{\{b\}\}\}$$, although Kuratowski's definition $$\{\{a\}, \{a, b\}\}$$ later superseded it.

It also appears that Principia Mathematica contained some form of ordered pair, perhaps as a primitive object or perhaps defined in terms of relations. (I actually can't tell based on the Wikipedia article.)

This answer on the philosophy stack exchange, however, explains that ordered pairs were used a long time ago to extend the expressive power of Aristotlean logic.

These devices have early traditional precursors, see Hodges, Traditional Logic, Modern Logic and Natural Language. E.g. Alexander of Aphrodisias and Ibn-Sina converted binary relational inferences into syllogisms by changing the domain of discourse to pairs.

As a brief, massively ahistorical aside, with two relations $$A(p, x)$$ and $$B(p, y)$$ defining projections out of a pair $$p$$ to $$x$$ and $$y$$ respectively we can build ordered pairs and then on to finite tuples of the form $$(a_1, (a_2, a_3))$$ or such. So a system that can handle binary relations could in principle be as expressive as first order logic.

It's not clear to me, though, whether Ibn Sina would have had an explicit concept of an ordered pair or not. This account of Ibn Sina's extension to Aristotlean logic may be ahistorical (it's possible that he used binary predicates but not ordered pairs).

If he did, I also don't know whether this would be the first usage of an ordered pair.

• For Principia, see $*21$ General Theory of Relations: "a relation may be regarded as the class of couples $(x,y)$ for which some given function $\psi(x,y)$ is true." Jun 20 at 12:54
• We may consider also Leibniz's approach to relations (see e.g. Massimo Mugnai, A Systematical Approach to Leibniz's Theory of Relations (1990)) as well as that of Jungius. Jun 20 at 13:03
• Related is the the following, but I don't believe it discusses issues as early as you're asking about: The empty set, the singleton, and the ordered pair by Akihiro Kanamori (2003; another copy here). Jun 20 at 13:03
• "The hope of finding a first comes to grief... If we describe a result with sufficient vagueness, there seems to be an endless sequence of those who had something within the vague specifications", May, Priority Chasing. Explicit use of objectified pairs does not predate Hamilton and Dedekind. What Hodges describes is just a retelling of Ibn Sina using modern conceptions. And one can retell already Euclid's ratios as pairs in a similar way. Jun 21 at 5:19
• Euclid, Book V, Def. 3:"A ratio is a sort of relation in respect of size between two magnitudes of the same kind." Joyce's commentary under it:"A ratio is a pair of magnitudes of the same kind considered as a pair, but soon identified with other ratios." Jun 21 at 6:49